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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Images of the Brownian sheet

Authors: Davar Khoshnevisan and Yimin Xiao
Journal: Trans. Amer. Math. Soc. 359 (2007), 3125-3151
MSC (2000): Primary 60G15, 60G17, 28A80
Published electronically: February 14, 2007
MathSciNet review: 2299449
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Abstract: An $ N$-parameter Brownian sheet in $ \mathbf{R}^d$ maps a non-random compact set $ F$ in $ \mathbf{R}^N_+$ to the random compact set $ B(F)$ in $ \mathbf{R}^d$. We prove two results on the image-set $ B(F)$:

(1) It has positive $ d$-dimensional Lebesgue measure if and only if $ F$ has positive $ \frac d 2$-dimensional capacity. This generalizes greatly the earlier works of J. Hawkes  (1977), J.-P. Kahane  (1985), and Khoshnevisan (1999).

(2) If $ \dim_{_\mathcal{H}}F > \frac d 2$, then with probability one, we can find a finite number of points $ \zeta_1,\ldots,\zeta_m\in\mathbf{R}^d$ such that for any rotation matrix $ \theta$ that leaves $ F$ in $ \mathbf{R}^N_+$, one of the $ \zeta_i$'s is interior to $ B(\theta F)$. In particular, $ B(F)$ has interior-points a.s. This verifies a conjecture of T. S. Mountford  (1989).

This paper contains two novel ideas: To prove (1), we introduce and analyze a family of bridged sheets. Item (2) is proved by developing a notion of ``sectorial local-non-determinism (LND).'' Both ideas may be of independent interest.

We showcase sectorial LND further by exhibiting some arithmetic properties of standard Brownian motion; this completes the work initiated by Mountford (1988).

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Additional Information

Davar Khoshnevisan
Affiliation: Department of Mathematics, The University of Utah, 155 S. 1400 E., Salt Lake City, Utah 84112–0090

Yimin Xiao
Affiliation: Department of Statistics and Probability, A-413 Wells Hall, Michigan State University, East Lansing, Michigan 48824

Keywords: Brownian sheet, image, Bessel--Riesz capacity, Hausdorff dimension, interior-point
Received by editor(s): September 12, 2004
Received by editor(s) in revised form: April 21, 2005
Published electronically: February 14, 2007
Additional Notes: This research was supported by a generous grant from the National Science Foundation
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.